Parity of Smaller Elements of Primitive Pythagorean Triple

Theorem

Let $\left({x, y, z}\right)$ be a Pythagorean triple, that is, integers such that $x^2 + y^2 = z^2$.

Then $x$ and $y$ are of opposite parity.

Proof

From Smaller Elements of Pythagorean Triple not both Odd, $x$ and $y$ are not both odd.

Aiming for a contradiction, suppose $x$ and $y$ are both even.

Then by definition they have $2$ as a common divisor.

But from Elements of Primitive Pythagorean Triple are Pairwise Coprime this cannot be the case.

So by Proof by Contradiction, $x$ and $y$ cannot both be even.

It follows that $x$ and $y$ are of opposite parity.

$\blacksquare$